论文标题
探索静电循环周围的电场：矩形，体育场，椭圆形和结
Exploring the electric field around a loop of static charge: Rectangles, stadiums, ellipses, and knots
论文作者
论文摘要
我们在沿循环均匀分布的静态电荷的连续一维环路周围研究了电场。对于平面中的矩形或体育场形环，我们发现电场可以随着环的拉长而进行对称性的干草叉分叉；该字段可以具有一个或三个零，具体取决于循环的纵横比。对于三维空间中打结的电荷分布，我们以数值计算电场，并将结果与以前发表的有关带电结周围平衡点的数量的理论界限进行比较。我们的计算表明，以前的界限远非鲜明。数字还提出了所有带有五个或更少十字架的带电结的实际平衡点数量的猜想。此外，我们还提供了带电结周围的等电位表面的第一批图像，并将其拓扑转换可视化，因为电势的水平各不相同。
We study the electric field around a continuous one-dimensional loop of static charge, under the assumption that the charge is distributed uniformly along the loop. For rectangular or stadium-shaped loops in the plane, we find that the electric field can undergo a symmetry-breaking pitchfork bifurcation as the loop is elongated; the field can have either one or three zeros, depending on the loop's aspect ratio. For knotted charge distributions in three-dimensional space, we compute the electric field numerically and compare our results to previously published theoretical bounds on the number of equilibrium points around charged knots. Our computations reveal that the previous bounds are far from sharp. The numerics also suggest conjectures for the actual minimum number of equilibrium points for all charged knots with five or fewer crossings. In addition, we provide the first images of the equipotential surfaces around charged knots, and visualize their topological transitions as the level of the potential is varied.
